College
College of Arts & Sciences
Mentor Information
Jean-Francois Biasse, Dmytro Savchuk
Description
An error-correcting code is a mathematical structure that models the flow of information through a noisy communication channel and seeks to mitigate errors through redundancy. These codes have a wide range of uses spanning from satellite transmissions to cryptographically secure communication schemes. An automorphism of a code is a permutation of the coordinate positions that preserves the code, and these symmetries are useful in helping to answer questions of code equivalence, classification, and decoding. Our work focuses on implementing and testing algorithms for code automorphism computation using GAP and the GUAVA package. We present an efficient algorithm for generating all codewords of a desired Hamming weight based on existing solutions to the Information Set Decoding (ISD) problem. Our second contribution is the development of a bipartite implementation of Leon's algorithm that constructs a codeword incidence matrix and iteratively refines row and column partitions. This refinement progressively reduces the search space for coordinate permutations, enabling efficient identification of code automorphisms. Overall, our contributions address a gap in the available GAP/GUAVA workflow by extending Leon-style code automorphism computation beyond the binary setting and replacing exhaustive codeword enumeration with information-set decoding, yielding a more scalable and efficient framework for automorphism computations in code-based cryptography.
Open-Source Software for Computing Code Automorphisms
An error-correcting code is a mathematical structure that models the flow of information through a noisy communication channel and seeks to mitigate errors through redundancy. These codes have a wide range of uses spanning from satellite transmissions to cryptographically secure communication schemes. An automorphism of a code is a permutation of the coordinate positions that preserves the code, and these symmetries are useful in helping to answer questions of code equivalence, classification, and decoding. Our work focuses on implementing and testing algorithms for code automorphism computation using GAP and the GUAVA package. We present an efficient algorithm for generating all codewords of a desired Hamming weight based on existing solutions to the Information Set Decoding (ISD) problem. Our second contribution is the development of a bipartite implementation of Leon's algorithm that constructs a codeword incidence matrix and iteratively refines row and column partitions. This refinement progressively reduces the search space for coordinate permutations, enabling efficient identification of code automorphisms. Overall, our contributions address a gap in the available GAP/GUAVA workflow by extending Leon-style code automorphism computation beyond the binary setting and replacing exhaustive codeword enumeration with information-set decoding, yielding a more scalable and efficient framework for automorphism computations in code-based cryptography.
